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Duality for Finite Gelfand Pairs

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Duality for Finite Gelfand Pairs Duality for finite Gelfand pairs M.V. Movshev∗ Mathematics Department, Stony Brook University, USA and ICTP South American Institute for Fundamental Research, IFT-UNESP, S˜ao Paulo State University, Brazil [email protected] June 26, 2021 Abstract Let G be a split reductive group, K be a non-Archimedean local field, and O be its ring of integers. Satake isomorphism identifies the algebra of compactly supported invari- G(O) ants Cc[G(K)/G(O))] with a complexification of the algebra of characters of finite- L dimensional representations (GL(C))G (C) of the Langlands dual group. In this note we O report on the results of the study of analogues of such an isomorphism for finite groups. In our setup we replaced Gelfand pair G(O) G(K) by a finite pair H G. It is con- ⊂ L ⊂ venient to rewrite the character side of the isomorphism as (GL(C))G (C) = ((GL(C) L O O × L L G (C) L L L arXiv:1707.03862v2 [math.RT] 22 Jul 2017 G (C))/G (C)) . We replace diagonal Gelfand pair G (C) G (C) G (C) by a dual ⊂ × finite pair Hˇ Gˇ and use Satake isomorphism as a defining property of the duality. In ⊂ this text we make a preliminary study of such duality and compute a number of nontrivial examples of dual pairs (H, G) and (H,ˇ Gˇ). We discuss a possible relation of our constructions to String Topology. ∗ The author would like to thank ICTP-SAIFR and FAPESP grants 2016/07944-2 and 2016/01343-7 for partial financial support. 1 Contents 1 Introduction 3 2 Duality for group pairs 11 2.1 Dualityinabstractterms . ..... 11 2.2 The matrix C(H, G) .................................. 15 2.3 Zonalsphericalfunction . ...... 17 2.4 Computing matrix C .................................. 19 2.5 Arithmetic properties of C ............................... 21 2.6 Dualityisomorphism .............................. 22 2.7 Galois group of C andduality............................. 24 2.8 MapsbetweenGelfandpair . 25 3 Examples 27 3.1 (Sn−1,Sn)........................................ 27 3.2 The group Sn Sk .................................... 28 ≀ 3.3 Semidirect product with an abelian group . ......... 30 3.4 Nonexamples..................................... 32 3.5 Onclassificationofdualpairs . ....... 33 3.6 Somesporadicexamples . .. .. .. .. .. .. .. 35 4 Some open questions 35 4.1 On the relation to other dualities . ........ 35 4.2 Duality for Gelfand pairs of Lie groups and locally compactgroups . 36 4.3 Onthecategorification. ..... 37 4.4 On the sufficient conditions for existence of the dual pair .............. 37 4.5 OntheStringTopology ............................. 38 2 1 Introduction Langlands duality for reductive groups and Satake isomorphism are important concepts that are used in the formulation of Langlands conjectures. The author’s motivation for writing this paper is placing these concepts into a more general context with the hope that in the future it might reveal new perspectives on the classical Langlands correspondence. Let us recall (see [8] for an accessible introduction to Langlands theory) some basic defi- nitions needed to state Satake isomorphism. We write G for a split reductive group, K for a non-Archimedean local field, and O for its ring of integers. By following the idea mentioned in the G(O) abstract we are going to replace the linear space Cc[G(K)/G(O))] = Cc[G(O) G(K)/G(O))] \ of Hecke operators by C[H G/H] with finite G and H. As usual, a group in superscript \ V G signals for taking G-invariants. Similarly the space of complexified algebraic characters L [GL(C)]G (C) = [GL(C) GL(C) GL(C)/GL(C)]1 of the Langlands dual group GL(C) over O O \ × C in our setup has a finite analogue C[Hˇ G/ˇ Hˇ ]. In general, we don’t assume that Gˇ splits into \ a product just like GL(C) GL(C). × We would like to think about elements of C[H G/H] C[G] as H-bi invariant functions \ ⊂ on G: f(hgh′) = f(g), g G, h, h′ H. The linear space C[H G/H] has two unital algebra ∈ ∈ \ structures. The first product f g is a point-wise multiplication of functions. The second is · the convolution f g, which in general is noncommutative. Recall that a pair of finite groups × H G is a Gelfand pair iff f g is commutative. Roughly speaking, we want to study duality ⊂ × on Gelfand pairs H G Hˇ Gˇ (1) ⊂ ⇔ ⊂ which interchanges the products . (2) ·⇔× We don’t expect Hˇ Gˇ to exist for an arbitrary Gelfand pair H G. Neither do we hope that ⊂ ⊂ Hˇ Gˇ will be given by a functorial construction. A notable exception is H = 1 and G is ⊂ { } × abelian. Then Hˇ = 1 and Gˇ = HomZ(G, C ). { } 1GL(C) is embedded diagonally into GL(C)) × GL(C) 3 One of the reasons for studying such an amorphous structure (besides formal resemblance with Satake isomorphism) is a link it provides between combinatorics and algebra. We postpone, for now, the formal definition of duality. Instead we formulate its corollary, which will be proven in due time (see (53)). To state it we notice that the space X = G/H splits into a disjoint union X = oi of H-orbits (in this paper we assume that the map G Aut(X) → has no kernel). TheS space of functions C[X] as a regular G-representation decomposes into a direct sum of irreducible multiplicity-one representations C[X]= i Ti. By a(H, G), b(H, G) we denote the arrays #oi , dim Ti , respectively. One of the nice featuresL of (1) is that it swaps { } { } the sizes of orbits, which live on combinatorial side of the correspondence, and the dimensions of representations, which belong to algebraic side: a(H, G)= b(H,ˇ Gˇ), b(H, G)= a(H,ˇ Gˇ) and X = Xˇ . (3) | | | | An illustration It is amusing to see that (H, G) = (Z ,S ) and (H,ˇ Gˇ) = (S ,S S ) is an 4 4 3 3 × 3 example of pairs (H, G) = (H,ˇ Gˇ) which satisfy (3). In fact, it is the simplest example of a non 6∼ self-dual, noncommutative pairs in duality (1). In the second pair, S is embedded diagonally into S S . The orbits of S in S S /S 3 3 × 3 3 3 × 3 3 coincide with the conjugasy classes in S3. These are 1 , (1, 2), (1, 3), (2, 3) , (1, 2, 3), (1, 3, 2) . { } { } { } So a(S ,S S )= 1, 2, 3 . The trivial, sign, and two-dimensional representations exhaust the 3 3 × 3 { } set Irrep(S3) of irreducible representations of S3. By Peter-Weyl isomorphism C ⊠ [G] ∼= T T . (4) T ∈IrrepM (G) b(S ,S S )= 1, 1, 4 . 3 3 × 3 { } 3 The group of rotations of a cube in R is isomorphic to S4 (it coincides with the group of permutations of diagonals). Let X be the set of faces of the cube. The stabilizer of a face is Z Z Z isomorphic to 4. Thus X ∼= S4/ 4. Under 4 X breaks into a union of two one-point orbits and one four-point orbit. We infer that a(Z ,S )= 1, 1, 4 . Frobenius formula 4 4 { } G G HomH (ResH Ti, C) ∼= HomG(Ti,IndH C) = HomG(Ti, C[X]) = Vi 4 G relates Z4 invariants in Ti with the multiplicities Vi of Ti in C[X]. As usual, IndH C stands G for the G-representation induced from the trivial representation of subgroup H, and ResH T stands for the restriction of T from G to H. Thus all representations Ti that contain the Z4- invariant vectors are subrepresentations of C[X]. Among such is the tautological 3-dimensional representation twisted by sign(σ). In order to construct another subrepresentation of C[X] we notice that S4 can be mapped onto S3. Geometrically this homomorphism comes from the S4 action on pairs of opposite edges of the tetrahedron. Under this map a generator e of Z4 maps to a reflection in S3. Because of this, the two-dimensional representation of S3 pulled back to S4 contains a Z4-invariant vector. Constants define a trivial subrepresentation in C[X]. Note that dim C + dim C2 + dim C3 = dim C[X] = 6. We conclude that b(Z ,S )= 1, 2, 3 . We see 4 4 { } that a(S ,S S )= b(Z ,S ), b(S ,S S )= a(Z ,S ). 3 3 × 3 4 4 3 3 × 3 4 4 For more examples the reader can consult Section 3. Algebras with two multiplicative structures Linear spaces C[H G/H] have more struc- \ tures then multiplications and mentioned above. We are going to enhance C[H G/H] by · × \ these structures and put the resulting object inside of a certain category. This will let us state in more precise terms what it means to swap the products in formula (2). We will be writing X for G/H. We start with an observation that the space C[X X]G is isomorphic to C[X]H = C[H G/H] : (5) × ∼ \ r : f(x,x′) f(x,x′ ), with St(x ) equal to H. The two products , on C[H G/H] and other → 0 0 × · \ structures are easier to describe in terms of C[X X]G. This what we are going to do now. × To this end, we start in a greater generality by fixing finite sets X,Y , which at the moment have no relation to the pair (H, G). C[X Y ] is an algebra with respect to the point-wise × multiplication of functions (f g)(x,y) := f(x,y)g(x,y) with the unit 1 (x,y) = 1.
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